yes, you should just study general music/ sound engineering/ audio design.
the skills you need are the same, only going a more general route (even if the option to specialise is open to you) gives you more options later on.
there is not a lot of work in audio for games, and it seems to be the one area where supply outstrips demand, so dont bet your livelyhood on it alone.

on the other hand if you prove to be really really good at what you do, and really dedicated, you will stand equil to better chance of getting a job in audio design in a games company to someone who did a specialised "audio for games" course.